Research And Application On Fourier's Convolution Type Quasi-Variational Principles In Non-Conservative System

Time domain analysis and frequency domain analysis for the structure's dynamic response which are the equivalent description of the same theory can be combined by the Fourier transform. Consequently the Fourier transform is a mathematical tool playing an important role in practical engineering applications. Non-conservative system consists of many kinds of disciplines.Especially in non-conservative elasto-dynamics system, the structure dynamic stability acted under non-conservative forces get much attention from researchers with the development of the technology. In the paper, based on the theory of the Fourier transform, Fourier's convolution type quasi-variational principles was presented by the variational integral method.Firstly,the idea to generalize a kind of convolution type variational principles was presented based on the Fourier transform. The feasibility to establish Fourier's convolution type variational principles was shown by introducing some examples.Secondly, Fourier's convolution type quasi-potential energy variational principles and quasi-complementary energy principles of initial value problems in non-conservative analytical dynamics were established. The first and second Fourier's convolution type generalized quasi-potential energy principles and quasi-complementary energy principles with two kinds and three kinds of variables were established. Quasi-stationary value conditions of each kind of Fourier's convolution type generalized quasi-variational principles in phase space and original space were derived to check presented variational principles.It is shown that these two kinds of variational principles are two different expressions in different space for the same variational principles.They are in inner-invertible relations.The responses of single-degree-of-freedom vibration system, two-degree-of-freedom vibration system and multi-degree-of-freedom vibration system were studied and their solutions were given.Thirdly, Fourier's convolution type quasi-potential energy variational principles and quasi-complementary energy principles of initial value problems in non-conservative elastodynamics were established by the variational integral method. The first and second type Fourier's convolution type generalized quasi-potential energy principles and quasi-complementary energy principles with two kinds and three kinds of variables were established. Fourier's convolution type generalized quasi-variational principles which indicate constitutive relationship(complementary strain energy or momentum constitutive relationship)and dynamic equilibrium equations were obtained. Fourier's convolution type generalized quasi-variational principles which indicate constitutive relationships (strain energy or momentum constitutive relationship) and geometric conditions were given. Also Fourier's convolution type generalized quasi-variational principles which indicate complementary strain energy and velocity constitutive relationship were presented. The example was presented to introduce the wave propagation characteristics of the quasi-stationary value conditions.Fourthly, using the variational integral method, Fourier's convolution type quasi-potential energy variational principles and quasi-complementary energy principles about natural frequency in generalized non-conservative elastodynamics system were established. The first and second type Fourier's convolution type generalized quasi-potential energy principles and quasi-complementary energy principles with two kinds and three kinds of variables were established. Expressions of corresponding variational principles in original space were obtained by Fourier inverse transform. The dynamic stability of the leipholz bar and the thin panel had been anylised by using Fourier's convolution type variational principles. It is shown that Fourier's convolution type variational principles can be used to the approximate calculation in engineering.